Solve {l}{-x^{2}+3x^{3}-3x+5x^{3}+2x^{2}}{4a^{2}-2b^2-2ab+(2a^2-b^2+ab)+5ab+b^2-2a}

$$\left. \begin{array} { l } { - x ^ { 2 } + 3 x ^ { 3 } - 3 x + 5 x ^ { 3 } + 2 x ^ { 2 } } \\ { 4 a ^ { 2 } - 2 b ^ { 2 } - 2 a b + ( 2 a ^ { 2 } - b ^ { 2 } + a b ) + 5 a b + b ^ { 2 } - 2 a } \end{array} \right.$$

Least Common Multiple

$$2x\left(-16abx^{2}+8ax^{2}+xb^{2}-2abx-3xa^{2}+ax+8\left(bx\right)^{2}-24\left(ax\right)^{2}+6ab+9a^{2}-3b^{2}-3a\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$x\left(x+8x^{2}-3\right)=8x\left(x-\left(-\frac{1}{16}\sqrt{97}-\frac{1}{16}\right)\right)\left(x-\left(\frac{1}{16}\sqrt{97}-\frac{1}{16}\right)\right) 6a^{2}-2b^{2}+4ab-2a=2\left(3a^{2}+2ab-a-b^{2}\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$2x\left(-16abx^{2}+8b^{2}x^{2}-24a^{2}x^{2}+8ax^{2}+xb^{2}-2abx-3xa^{2}+ax+6ab+9a^{2}-3b^{2}-3a\right)$$

Expand the expression.

$$16b^{2}x^{3}-32abx^{3}-48a^{2}x^{3}+16ax^{3}+2b^{2}x^{2}-4abx^{2}-6a^{2}x^{2}+2ax^{2}+12abx+18xa^{2}-6xb^{2}-6ax$$

Evaluate

$$x\left(8x^{2}+x-3\right),\ 2\left(2ab+3a^{2}-b^{2}-a\right)$$